Research on ideal class groups of algebraic number fields and number theoretic functions and its applications

本研究的目的是进一步发展代数数域的理想类群的基本理论，特别是考虑到Iwasawa理论。虽然我们在研究的第一年没有发现p-二义理想类群的理论行为，但我们验证了我们对p=3的Iwasawa不变量都为零的实二次场的密度估计非常接近于通过修改Cohen-Lenstra启发式对类数得到的猜想比。与已知的估计与推测的估计相去甚远的情况相比，我们的分析是有趣的。明年，我们将证明，对于p=2的Iwasawa不变量都为零的实二次域的密度为零。这也很有趣，因为根据我们之前的结果，p=3的密度大于0.7，而p>3的密度被推测为正的。从那时起到本研究的最后一年，我们有机会与山本根博士（东京电机大学）共同研究，并利用属理论和中心扩展理论确定了p=2的Iwasawa不变量均为零的所有实阿贝尔2-扩展域。本研究成果已在韩日数论研讨会和ICM2006等会议上发表。虽然我们在数论和组合理论的相互应用方面没有取得令人满意的结果，但我们通过组织两领域研究人员聚集的小型会议，为提出和解决这两个领域的共同问题发挥了重要作用

The purpose of this research is to develop the fundamental theory of ideal class groups of algebraic number fields further, especially taking account of Iwasawa theory. Though we did not find out a theoretical behavior of p-ambiguous ideal class groups in the first year of this research, we verified that our estimate on the density of real quadratic fields whose Iwasawa invariants for p=3 are all zero is very near to the conjectural ratio obtained by modifying Cohen-Lenstra Heuristics on class numbers. As compared with the case p>3 in which known estimates are too far from the conjectural one, our analysis is interesting. Next year, we showed that the density of real quadratic fields whose Iwasawa invariants for p=2 are all zero is zero. This is also interesting, because the density for p=3 is more than 0.7 by our previous result and the one for p>3 is conjectured to be positive. From that time to the last year of this research, we had an opportunity to have a joint research with Dr. Gen Yamamoto (Tokyo Denki University) and determined all real abelian 2-extension fields whose Iwasawa invariants for p=2 are all zero, by using genus theory and the theory of central extensions. The results obtained in this research are presented in some of conferences like as Korea-Japan Number Theory Seminar and ICM2006. Though we did not get a satisfying result about mutual application between number theory and combinatorial theory unfortunately, we played an important role in posing and solving common problems in both fields by organizing mini conferences where researchers in both fields gathered